Student Academic Learning Services Page 1 of 6 Laws about gases Charles’ law Volume is directly proportional to temperature. V = cT, where c > 0 is constant. French balloonist Jacque Charles noticed that if air in a balloon is Charles: Heat a balloon heated, the balloon expands. For an ideal gas, the relationship between and it gets bigger! V and T is linear (as long as pressure is constant). Boyle’s law Pressure is inversely proportional to volume.
, where a > 0 is a constant.
Robert Boyle noticed that when the volume of a container holding an amount of gas is increased, pressure decreases, and vice versa (while the temperature is held constant). Note that this is not a linear Boyle: Relieve pressure relationship between p and V. on yourself by wearing puffy clothes. Combined gas law
Mathematically, you can combine the Charles’ and Boyle’s laws to get , where k is a
constant. Ideal gas law This law combines the relationships between p, V, T and mass, and gives a number to the constant! The ideal gas law is: pV = nRT, where n is the number of moles, and R is universal gas constant. The value of R depends on the units involved, but is usually stated with S.I. units as: R = 8.314 J/mol∙K. Ideal gas law for air For air, one mole is 28.97 g (=0.02897 kg), so we can do a unit conversion from moles to kilograms.
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This means that for air, you can use the value R = 287 J/kg∙K. If you use this value of R, then technically the formula should be written as pV = mRT, where m represents the mass of air in kg (and we avoid having to do any calculations with moles.)
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Student Academic Learning Services Page 2 of 6 Relevant concepts for gases Pressure
Pressure is defined as a force applied over the surface area of an object. So,
For example, picture a person standing on a box that sits on the floor. The person weighs 800 N and the box, viewed from above, is the shape of a square with a side length of 1 meter. The pressure of the box on the floor would be:
Pressure is measured in Pascals (Pa), and 1 Pascal is defined as 1 N/m2. Therefore, the pressure on the floor is 800 Pa.
Another common unit for pressure is pounds per square inch (psi). Barometric Pressure Also known as atmospheric pressure, this is pressure applied to any object that is in an atmosphere (e.g. on earth). Standard atmospheric pressure, at sea level, at a temperature of 25 °C, is 101.325 kPa. Barometric pressure is measured with a barometer. Manometric Pressure Also known as the gauge pressure, this is the internal pressure of the system, and does not include the barometric pressure. Manometric pressure is measured with a manometer. Note that this value can actually be negative (see below). Absolute Pressure This is the total amount of pressure including both the manometric and barometric pressures. You can use the formula: Absolute Pressure = Barometric Pressure + Manometric Pressure With this formula, it is important to note that manometric pressure can, and will, have a negative value whenever the absolute pressure of the system is lower than the barometric pressure. When this happens, you have a partial vacuum. Specific Volume The specific volume of an object is defined as its volume (i.e. space occupied) per unit mass. The symbol is a lower case v (which is sometimes written in italics), and the formula can be written as:
where capital is the total volume.
The normal unit of measurement for specific volume is m3/kg. Specific volume is also the
reciprocal of density: , where (Greek letter rho) is the density of an object (or substance).
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Student Academic Learning Services Page 3 of 6 Temperature Temperature is a less straightforward concept than pressure. Temperature is a measure of the amount of thermal energy that exists in a physical space, relative to a zero point of energy, which we define as absolute zero. In science, temperature is usually measured on the Kelvin scale, where 0 K is absolute zero, and 273.15 K is the freezing point of water (273.15 K = 0 °C). Problems Gas law problems 1) The pressure-volume relationship for a given gas is assumed to be Pv1.2 = Constant. At a pressure of 1.5 bars, the volume is of the gas is 55 mL. The mass of the gas inside is 80 mg. If temperature is kept constant, what should the new pressure be (in bars) when the volume is decreased to 35 mL? 2) Calculate the mass in grams of air contained in a spherical balloon with a diameter of 5 m. The temperature of the air inside the balloon is 55.0 °C and the manometric pressure is measured at 1.5 bars. 3) Calculate the volume of 1.2 kg an ideal gas with a molar mass of 55 g/mol at a pressure of 120 kPa, and a temperature of 80°C. Also, what is the specific volume and the density of the gas? Piston pressure problem 4) In a piston system, the piston diameter is 25.0 mm, and the mass of the piston and platform is 55.0 g. If a 0.584 kg weight is placed on top of the platform to equilibrate the pressure, then what is the manometric pressure inside? Assume the barometric pressure is 101.300 kPa and g = 9.807 m/s2.
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Student Academic Learning Services Page 4 of 6 Answers Gas Laws Question 1 Given: Equation:
( )( )
( ) ( )( )
( )
Note that we didn’t need to know that the mass was to solve this problem – i.e. the mass (80 mg) was extraneous information. We also didn’t need to convert from mL to m3 because the mL cancelled out in the equation on the right (because it is a proportion). Therefore, the new pressure, after compression, is 2.58 bar. Question 2 Given: Equations: (using the value of R for air)
( )
( )( )
( ) ( ) ( )
Therefore, the mass of the air in the balloon is 834000 g.
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Student Academic Learning Services Page 5 of 6 Question 3 Given: Equations:
( ) ( ) ( )
Specific Volume Density
⁄ ⁄
Therefore the volume of the gas is 0.534 m3, the specific volume is 0.455 m3/kg and the density is 2.25 kg/m3. Piston Pressure Problem Question 4 There are three forces pushing the piston downward:
Fbar = The force of the barometric pressure
Fapplied = The force of the applied weight
Fcyl = The force of the weight of the piston and platform. There is only one force pushing upward:
Fmano = The force of the manometric pressure inside the piston Since the system is in equilibrium (i.e. it isn’t moving), the sum of the forces acting downward must be equal to the forces acting upward.
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Student Academic Learning Services Page 6 of 6
Since Force of weight = mass × gravitational acceleration And Force of pressure = pressure × area All of the quantities on the left side of this equation can be determined from the given information, but we have to be careful to use the correct units: kg, m, and Pa.
( m) m2 By converting the given diameter of 25 mm to 0.025 m and dividing by 2 to get the radius.
mapplied = 0.584 kg This is already in the correct units
mcylinder = 55.0 g = 0.055 kg
pbar = 101300 Pa g = 9.807 m/s2
Substituting all of this into the equation for pmano above, we get: a m2 m/s2 kg m/s2
m2 All three of the multiplications on the type should result in N as the units, since they are the three forces acting downward from the original equation. The result is:
m2
m2 2 /m a A Pascal is a Newton per square metre. Therefore, the manometric pressure is about 114000 Pa or 114 kPa.
Images: 1) http://www.bbk.ac.uk/boyle/Issue4.html 2) Library of Congress digital ID ppmsca.02185
www.durhamcollege.ca/sals Student Services Building (SSB), Room 204 905.721.2000 ext. 2491 This document last updated: 7/29/2013